Quantum Speed-ups for Semidefinite Programming

TL;DR

This paper introduces a quantum algorithm that significantly accelerates solving semidefinite programs, achieving a square-root speed-up over classical methods and establishing fundamental lower bounds.

Contribution

The paper presents the first quantum algorithm for SDPs with proven optimality bounds, combining quantum Gibbs sampling and the multiplicative weight method.

Findings

Achieves a worst-case running time with square-root speed-up over classical algorithms.

Proves a lower bound indicating the algorithm's near-optimality in terms of problem size.

Modifies classical SDP algorithms to remove the need for solving inner linear programs.

Abstract

We give a quantum algorithm for solving semidefinite programs (SDPs). It has worst-case running time $n^{\frac{1}{2}} m^{\frac{1}{2}} s^2 \text{poly}(\log(n), \log(m), R, r, 1/\delta)$, with $n$ and $s$ the dimension and row-sparsity of the input matrices, respectively, $m$ the number of constraints, $\delta$ the accuracy of the solution, and $R, r$ a upper bounds on the size of the optimal primal and dual solutions. This gives a square-root unconditional speed-up over any classical method for solving SDPs both in $n$ and $m$. We prove the algorithm cannot be substantially improved (in terms of $n$ and $m$) giving a $\Omega(n^{\frac{1}{2}}+m^{\frac{1}{2}})$ quantum lower bound for solving semidefinite programs with constant $s, R, r$ and $\delta$. The quantum algorithm is constructed by a combination of quantum Gibbs sampling and the multiplicative weight method. In particular it is based on a classical algorithm of Arora and Kale for approximately solving SDPs. We present a modification of their algorithm to eliminate the need for solving an inner linear program which may be of independent interest.